Abstract

We report on the challenges and limitations of direct coupling of the magnetic field from a circuit resonator to an electron spin bound to a donor potential. We propose a device consisting of a lumped-element superconducting resonator and a single donor implanted in enriched 28Si. The resonator, in contrast to coplanar waveguide resonators, includes a nano-scale spiral inductor to spatially focus the magnetic field from the photons within. The design promises approximately two orders of magnitude increase in the local magnetic field, and thus the spin to photon coupling rate g, compared to the estimated coupling rate to coplanar transmission-line resonators. We show that by using niobium (aluminum) as the resonator’s superconductor and a single phosphorous (bismuth) atom as the donor, a coupling rate of g/2π=0.24 MHz (0.39 MHz) can be achieved in the small photon limit. For this truly linear cavity quantum electrodynamic system, such enhancement in g is sufficient to enter the strong coupling regime.

pacs:

Silicon-based spin qubits, including gate-defined quantum dot Hanson et al. (2007); Loss and DiVincenzo (1998); Petta et al. (2005) and single-atom Zwanenburg et al. (2013); Pla et al. (2012) devices, use the spin degree of freedom to store and process quantum information, and are promising candidates for future quantum electronic circuits. The electronic or nuclear spin is well decoupled from the noisy environment, resulting in extremely long coherence times Saeedi et al. (2013); Tyryshkin et al. (2012); Steger et al. (2012) desirable for fault-tolerant quantum computing. Single-atom spin qubits offer additional advantages over quantum dot qubits such as longer coherence due to strong confinement potentials, and are expected to have better reproducibility by nature. Therefore, it is no surprise that silicon-based single-atom spin qubits hold the record coherence times of any solid state single qubit Muhonen et al. (2014). However, this attractive isolation from sources of decoherence comes at the price of relatively poor coupling to the control and readout units. This causes relatively long qubit initialization times, degraded readout fidelity, and weak spin-spin coupling for multi-qubit gate operations Awschalom et al. (2013).

One simple way to enhance the coupling rate is to increase the ac magnetic field from the external circuit. In this regard, superconducting circuit resonators are attractive due to their relatively large quality factors, ease of coupling to other circuits, their capability of generating ac magnetic fields by carrying relatively large currents, and monolithic integration with semiconductor devices.

For many years, superconducting microwave resonators have had extensive applications that range from superconducting qubit manipulation Wallraff et al. (2004) and inter-qubit coupling Majer et al. (2007) to dielectric characterization Sarabi et al. (2016). One of the most commonly used superconducting quantum computing architectures is one based on cavity quantum electrodynamics (cQED) Blais et al. (2004), in which a 2D (circuit based) or 3D cavity is employed to initialize, manipulate and readout the superconducting qubit. Superconducting circuit cavities have not yet found a similar prevalence in spin qubit circuits due to the fact that the magnetic field of a typical superconducting resonator and the spin magnetic moment are relatively small, leading to an insufficient spin-photon coupling strength for practical purposes. The direct magnetic coupling of coplanar resonators to donor electrons in silicon Eichler et al. (2017); Imamoğlu (2009); Tosi et al. (2014) and diamond nitrogen-vacancy centers Kubo et al. (2010); Amsüss et al. (2011) can only achieve a maximum single-spin coupling rate of a few kHz.

Several methods have been proposed to enhance the coupling of a single spin to a photon within a superconducting circuit resonator. It is easier to couple a photon to quantum dot spin qubits than to single-atom spins because in the former, the spin dynamics can be translated into an electric dipole interacting with resonator’s electric field Petersson et al. (2012); Viennot et al. (2015); Hu et al. (2012). Such architectures, however, require hybridization of the spin states with charge states, coupling charge noise to the spin and thus affecting its coherence. For the single-atom spin qubits, indirect coupling to a resonator via superconducting qubits Kubo et al. (2011); Twamley and Barrett (2010) can enhance the coupling, but imposes nonlinearity on the circuit complicating cQED analysis, and also introduces loss from Josephson junction tunnel barriers Simmonds et al. (2004); Martinis et al. (2005) and magnetic flux noise Kumar et al. (2016).

Figure 1: (a) Schematic of the circuit showing the resonator galvanically coupled to the coplanar waveguide (CPW). CGND is the resonator’s capacitance to ground. Pin, Pout, Lg and Lp show the microwave input power, output power, geometric inductance and the parasitic inductance of the resonator circuit, respectively. This arrangement of Lg and Lp is valid in the limit of C≫Cc,CGND. (b) Layout of the device showing resonator’s capacitor C and inductor L. Blue and orange show the top and bottom superconducting layers, respectively. The area within the red dashed square is magnified. The spiral inductor within the green dashed square is further magnified to show (c) the single spiral geometry (red dot represents the spin). Dimensions are w=s=t=din/2=30 nm, compatible with standard electron beam lithography, and d0=ds=25 nm. The crystalline 28Si (x-28Si) growth is required only in the volume surrounding the spin, and amorphous or polycrystalline Si is acceptable everywhere else. The relatively weak magnetic field Bvia (directions shown represent magnetic fields at the location of the spin) from the via determines the best choice for the direction of the static magnetic field, B0∥(Bvia×Bac), where Bac is the magnetic field from the spiral inductor. This ensures that the total ac field is perpendicular to B0. Current within the spiral is shown by green in- or out of plane vectors.

In this paper, we show that replacing the coplanar transmission line resonator with a lumped-element circuit resonator that includes a spiral inductor, can lead to a dramatic enhancement in the spin-photon magnetic coupling rate g of approximately two orders of magnitude. As we will discuss, this improvement is a result of employing a lumped-element trilayer design along with nano-scale spiral loops which effectively localize the resonator’s magnetic field at the location of the spin, eliminating the need for an Oersted line Laucht et al. (2015). We also show that this coupling rate is enough to take the spin-resonator hybrid system to the strong coupling regime, where g is larger than or approximately equal to the resonator decay rate κ. Due to the relatively small total inductance Ltot, large capacitance C and hence small impedance Z=√Ltot/C of the device, coplanar interdigitated capacitances are insufficient to achieve the desired operation frequency ω0=1/√LtotC. Therefore, the proposed device geometry, compatible with standard micro- and nanofabrication techniques, includes a trilayer (parallel-plate) capacitor with a deposited insulating layer. Trilayer capacitors, however, give rise to a lower resonator quality factor Q than that of a typical coplanar geometry. Nevertheless, as our calculations show, the increase in g resulted from employing a trilayer lumped-element design is large enough to overcome the limited Q, allowing for strong spin-photon coupling.

The lossless dynamics of a spin-12 system with spin transition frequency ωs coupled with rate g to a cavity resonant at ω0, is described by the Jaynes-Cummings Hamiltonian, H=ℏω0(^a†^a+1/2)+(1/2)ℏωs^σz+ℏg(^a†^σ−+^σ+^a)Jaynes and Cummings (1963). Here, ^a and ^σ are photon and spin operators, respectively. The spin-photon magnetic coupling rate is obtained as Extra close brace or missing open brace where ge≃2 is the electron g-factor, μB is the Bohr magneton, and Extra close brace or missing open brace is the root mean square (RMS) local magnetic field at the location of the spin with n=0 photons on resonance. |g⟩ and |e⟩ are the ground and excited spin states, respectively, that couple to the microwave field. Finally, the spin rotation speed can be enhanced with a larger local RMS magnetic field Extra close brace or missing open brace for any arbitrary n.

The schematic and layout of the proposed circuit are shown in Figs. 1(a) and 1(b-c), respectively, where a LC resonator is coupled to a coplanar waveguide (CPW) through direct (galvanic) connection through a coupling inductor Lc and the donor is within the resonator’s deliberate inductor Lg. The galvanic coupling, employed in previous experiments Vissers et al. (2015), can help to achieve the desired CPW-resonator coupling rates especially when the resonator impedance is significantly different from the CPW’s characteristic impedance Z0=50Ω. The capacitance C is provided using a trilayer capacitor. The RMS current through the inductor at an average photon number ¯n on resonance is Iac=√(¯n+1/2)ℏω0/Ltot, where ω0 is the resonance frequency and Ltot denotes the total inductance within the resonator circuit. Since the desired ac magnetic field from the spiral loop(s) is proportional to Iac, it is clear that the inductance (or impedance Z) must be minimized and photon frequency must be maximized for the maximum magnetic field. Therefore, it is necessary to study the sources of inductance and obtain operation frequency limitations.

We study two sets of materials, one that uses niobium as superconductor and phosphorous as the donor (Nb/P), and one with aluminum as superconductor and bismuth as donor (Al/Bi). Fabrication is considered to be slightly better established with Al, whereas Nb has a much higher critical magnetic field of Bc1,Nb=0.2 T, which is the highest Bc1 among the single-element superconductors. Through the Zeeman effect, for the Nb/P set, the uncoupled electron spin-up (|↑⟩) and spin-down (|↓⟩) states are split by the photon frequency ω0/2π=5.6 GHz corresponding to B0=ℏω0/geμB≃Bc1,Nb. For the Al/Bi set, we consider operating at B0<10 mT (below the critical magnetic field of Al), and use the splitting of the spin multiplets (ω0/2π=7.375 GHz) of the Bi donor. These splittings arise from the strong hyperfine interaction between the electron (S=1/2) and Bi nuclear spin (I=9/2) Wolfowicz et al. (2012), leading to a total spin F and its projection mF along B0. By employing the |F,mF⟩=|5,−5⟩↔|4,−4⟩ transition corresponding to the largest ^Sx matrix element (0.47), and a static magnetic field of B0=5 mT, the multiplet degeneracy at B0=0 is lifted by more than 20 MHz, enough to decouple the nearby transitions Bienfait et al. (2016); Mohammady et al. (2010).

For both Nb/P and Al/Bi sets, Ltot is simulated and shown in Fig. 2(a) as a function of number of spiral loops, Nloops (see supplemental material sup () for details of inductance calculations and simulations). Increasing Nloops from 1 to 2 increases Bac,0, but, for Nloops>2, the competing effect of larger Ltot due to larger loop radii suppresses Iac and lowers g. This effect is clearly demonstrated in Fig. 2(b) as the optimum Nloops=2 for both material sets, where the vacuum fluctuation’s coupling rates for the Al/Bi and Nb/P configurations are obtained as g/2π=0.26 MHz and 0.17 MHz, respectively. If we consider a double layer (2S) spiral inductor (see supplemental material sup ()), these coupling rates approach g/2π=0.39 MHz and 0.24 MHz, respectively. To the knowledge of the authors, these values are approximately two orders of magnitude larger than a previously proposed architecture Tosi et al. (2014). As we report in the supplemental part sup (), this new regime of g can lead to significant improvements in spin qubit manipulation rates and readout fidelities, and potentially enables new experiments such as individual spin spectroscopy.

We now discuss a relatively simple proof-of-principle experiment. We consider a low-impedance resonator on top of a Bi-doped substrate or 28Si layer and propose to measure the spectrum in the small photon limit while the electron spin transitions are tuned across the resonator bandwidth. To further simplify this experiment, a single spiral loop can be employed with dimensions compatible with standard photolithography. In this simplified experiment with no single donor implantation or e-beam lithography requirements, one can choose Al as superconductor and thus the kinetic inductance within the circuit is negligible with micron-scale dimensions of the inductor loop. When Ns spins are incorporated and exposed to the relatively uniform magnetic field inside the inductor loop, the collective coupling rate from the spin ensemble becomes gcol=g√NsImamoğlu (2009). This collective coupling rate would yield more clear spin-resonator interaction features in the spectra for this first proof-of-principle version of the device compared to the single donor device. In our proposed device, tuning of the spin(s) can be performed using magnetic (Zeeman) or electric (Stark, see Fig. 1) fields, or a combination of both.

Since the spin ensemble spectroscopy device is micron-scale, high-frequency simulations using a electromagnetic simulation software become feasible. The layout of the device is shown in Fig. 3(a). We consider galvanic coupling to the CPW and assume a resonator internal quality factor of Qi=104. The simulated transmission S21=Vout/Vin through the CPW shows a characteristic resonance circle with diameter of 0.5. This indicates that, because of the galvanic coupling and despite Z≃0.4Ω≪Z0, critical coupling (Qe=Qi) can be achieved (see Fig. 3(b)).

From the high-frequency simulation, the geometric inductance of the loop is extracted as Ltot=8.5 pH. Despite the significantly larger inductor dimensions of the ensemble device, its impedance is similar to that of the nanoscale (single-atom) device for Nloops=2−3, primarily due to the smaller contribution of the kinetic inductance in the ensemble device. Therefore, we conclude that the CPW can also be sufficiently coupled to the single-atom device with a nanoscale spiral inductor.

Figure 3: (a) Layout of the simulated Al resonator coupled to an ensemble of Bi spins (ground plane not shown). Here, the dielectric thickness for the capacitor C is assumed to be d=50 nm. (b) Resonator transmission simulation around its resonance frequency f0=7.2 GHz. Spectroscopy simulation of the hybrid resonator-spin ensemble device near f0 for (c) zero spin-resonator detuning and (d) with tunable static magnetic field B0 near the degeneracy field Bd.

By assuming a uniform Bac inside the inductor loop that is applied to spins within 500 nm from the substrate surface with doping density of 1017 cm−3, we find gcol≃1 MHz using the method described in Ref. Bienfait et al. (2016b). We use a theoretical model previously developed for a different quantum device with similar physics to simulate the spectroscopy of the spin ensemble Sarabi (2014). Figures 3(c-d) show the spectroscopy simulation results using Q=Qi/2=Qe/2=5000 for the resonator. As done previously Sarabi et al. (2015), one can extract important parameters of this hybrid spin-resonator system directly from the spectroscopy data. These parameters include gcol, the resonator Q and the relaxation time T1 of the spin ensemble.

In summary, we have proposed and designed a novel device that enhances the coupling of a single atom spin to the magnetic field of a circuit resonator by approximately 100 times compared to the previously proposed architectures that use coplanar transmission line resonators. This dramatic improvement is a result of using a low impedance, lumped element resonator design and a spiral inductor geometry. We showed the possibility of entering the strong coupling regime necessary for practical purposes, i.e., spectroscopic measurements and qubit realization. As shown in the supplemental part sup () using the principles of cavity quantum electrodynamics, this large g can lead to a significantly enhanced spin relaxation rate desired for qubit initialization, tens of megahertz spin rotation speed during manipulation without the need for an Oersted line, and superb dispersive readout sensitivity. Moreover, this architecture can be useful for coupling distant qubits using cavity photons for the realization of multi-qubit gates.

“Prospective two orders of magnitude enhancement in direct magnetic coupling of a single-atom spin to a circuit resonator”
Prospective two orders of magnitude enhancement in direct magnetic coupling of a single-atom spin to a circuit resonator

Appendix A Inductance simulations and calculations

The total inductance Ltot=Lg+Lp within the resonator consists of the geometric inductance Lg of the spiral inductor giving rise to the magnetic field Bac that couples to the donor electron spin, and parasitic inductance Lp which does not contribute to Bac and only limits it. Lg consists of the trace inductance Lt arising from the length of the spiral trace, and some mutual inductance LM between the loops such that Lg=Lt+LM. The parasitic inductance Lp arises from the kinetic inductance of the spiral Lk, kinetic inductance within capacitor plates LC,k and the geometric self-inductance of the capacitor LC,g such that Lp=Lk+LC,k+LC,g. Since Lp does not create any magnetic fields at the location of the spin and only limits Iac through Ltot, we want to minimize it. Below, we describe our estimation of all of the aforementioned inductances.

In order to obtain an accurate estimation of Lg and Lp, we performed calculations as well as software simulations. We study two different device geometries, one using a single spiral inductor and the other using a double spiral (2S) inductor shown in Figs. 1(c) and S1, respectively. Nloops refers to the number of loops in a single spiral layer regardless of the geometry, e.g., Nloops=2 for both Figs. 1(c) and S1. In general, the single spiral design is expected to be easier to fabricate at the expense of smaller g compared to the double-spiral geometry. This results in four configurations under consideration, i.e. Al/Bi, Al/Bi-2S, Nb/P and Nb/P-2S.

Figure S1: Layout of the double-spiral (2S) device in the vicinity of the nanoscale spiral inductor. Here, ds=27.5 nm and other dimensions are identical to those in Fig. 1(c).

Lg is approximately calculated and also separately simulated. In the calculations, for simplicity, we assume that each loop of the spiral inductor is truly circular and estimate Lg using the geometric mean distance (GMD) method to the second order in the ratio of the conductor diameter to the loop radius Grover (2004). This independent loop approximation (ILA) ignores the mutual inductance LM within the spiral loops. However, one should note that LM contributes to Bac and is not parasitic. Nevertheless, in addition to the ILA, we also simulated the spiral geometry in FastHenry 3-D inductance extraction program Kamon et al. (1994), which accounts for the mutual inductances LM within the spiral geometry. Figure S1 shows a comparison between the simulation and the approximate calculation, where the latter ignores LM. Clearly, LM constitutes a larger portion of Lg with increasing Nloops, but is negligible up to Nloops≈2,3 where g is optimum (see Fig. 2(b)).
The total inductance Ltot in Fig. 2(a) is plotted using Lg from FastHenry simulations. However, for simplicity, the ac field contributed by LM was not taken into account in calculating g, resulting in an underestimated g in Fig. 2(b).

Figure S2: Comparison between the independent loop approximation (ILA) and FastHenry simulations of the spiral geometric inductance versus for different spiral loop counts Nloops for the single spiral and double spiral (2S) geometries.

The kinetic inductance Lk of the spiral loop is caused by the kinetic energy of the quasi-particles within the superconductor and hence does not create any magnetic fields. In our design, Lk is the largest part of Lp. Using the Cooper-pair density ns and mass mC for a particular superconducting material, the length of the superconducting line ls and its cross-sectional area As, one can estimate Lk=mCls/4nse2AsTinkham (1996), where e denotes the electron charge. For the single-layer spiral Nb/P resonator with 1≤Nloops≤5 we obtain 0.6pH≤Lk≤7pH, linearly proportional to the spiral trace length.

The kinetic inductance within the capacitor plates, LC,k≈60 fH, is negligible due to the large cross-sectional area of the plates. The geometric self-inductance of the capacitor, calculated using a stripline model, is LC,g≈62 fH. To confirm this, we simulated the resonator geometry including the capacitor and found LC,g=64 fH, in agreement with the calculated value and also negligible for Nloops≳2. The relatively small LC,g is due to the small capacitor insulating thickness d0.

In order to achieve the desired resonance frequencies with the relatively small inductances shown in Fig 2(a), relatively large capacitances are required. By using a capacitor insulator thickness of d0=25 nm, we can keep the square-shaped capacitor dimensions below 200 μm, and also suppress capacitor’s geometric inductance which contributes to Lp.

A simulation of g as a function of the spiral trace width w, spacing s and thickness t, was performed. By assuming w=s=t, the results showed that, for the Nb (Al) set, the optimum g is obtained when 30nm≲w=s=t≲35 nm (w=s=t≈20 nm), weakly depending on whether the 1S or the 2S geometry is used. If e-beam lithography resolution of 20 nm is implemented, the Al set can yield a spin-photon coupling rate of g/2π=0.45 MHz.

At the location of the spin, Bac is approximated as the sum of the magnetic field from all loops, i.e, Bac=∑loopsμ0Iac/2R′loop where R′loop≡(d2in+4d2s)3/2/2d2in is defined as a characteristic radius which accounts for the spin location with respect to the center of the loops and ds denotes the vertical spin displacement from this center (see Figs. 1(c) and S1). Note that the assumption of current flowing in the center of the spiral conductor underestimates Bac, Bac,0 and g, because in reality the majority of current will flow closer to the superconductor-silicon interface due to the relatively large electric permittivity (ϵr=12) of silicon.

For a better understanding of the dependence of g on the number Nloops of the spiral loops, a naive picture may be helpful to the reader. To the first order for Nloops=1, Iac∝1/√Lg+Lp, and Lg∝N2loopsln(Nloops), and Bac,0,g∝Iacln(Nloops), but Lp is a weaker function of Nloops than Lg. This suggest that by increasing Nloops, Bac,0 and hence g increase up to a point where Lg approaches Lp, and Iac begins to drop as Nα, with α<−1. Employing a lumped element design provides the required flexibility to reach this optimum Lg and the corresponding Nloops.

The condition for the system to enter the strong coupling regime in the small photon limit is Q≳ω0/g, where Q=ω0/κ and κ are resonator’s total quality factor and total photon decay rate, respectively. Thanks to the relatively large spin-photon coupling rate, resonator Q’s in the range of 104s are enough to take the system to the strong coupling regime for all four configurations (see Fig. S3(b)). In general, in the limit of low drive power and low temperature, trilayer resonators are significantly lossier than coplanar resonators due to the fact that almost all the photon electric energy is stored within the parallel-plate capacitor dielectric which contains atomic-scale defects. These defects act as lossy two-level fluctuators at small powers and low temperatures, and limit the resonator QMartinis et al. (2005). However, loss tangents as small as 10−5 have been measured for deposited amorphous hydrogenated silicon (a-Si:H) at single photon energies O‘Connell et al. (2008) promising resonator Q’s of approximately 105. More recently, elastic measurements have indicated the absence of tunneling states in a hydrogen-free amorphous silicon film suggesting the possibility of depositing “perfect” silicon Liu et al. (2014), promising even higher Q trilayer resonators using silicon as capacitor dielectric.

In Fig. S1, we see the following optimum combination of materials: i) Amorphous or polycrystalline Si can be deposited on the metal layers; since they have similar permittivities to crystalline Si, the capacitor size will be similar. ii) Since the spin does not have a metal layer below it, one can still deposit crystalline 28Si and then implant the donor, thus avoiding the fast decoherence which would result from the non-crystalline, non-enriched films Fuechsle et al. (2012). From the fabrication point of view, it is easiest to use the same 28Si film as the capacitor dielectric which, in general, will be in the polycrystalline form when deposited on the bottom capacitor plate (see Figs. 1(c) and S1). It is noteworthy that using a trilayer design makes the resonator quality factor Q independent from the surrounding material as almost all the electric field energy is confined within the capacitor dielectric. This is useful for the integration of single electron devices that require lossy oxide layers.

Appendix D Possibility of donor ionization

It has been previously shown that the donor can be ionized in the vicinity of metallic or other conductive structures due to energy band bendings Fuechsle et al. (2012). For Al-Si interface, the relatively small work function difference of -30 meV (4.08 eV for Al and 4.05 eV for Si) is smaller than the donor electron binding energy of -46 meV Jagannath et al. (1981), and hence using aluminum is expected to allow the donor bound state. However, the Nb work function of 4.3 eV causes significant band bending which can result in donor ionization. By biasing the microwave transmission line and hence the resonator with a DC voltage, one can modulate the potential energy arrangement in the neighborhood of the donor and recreate a bound state. If a conduction band electron is required to fill the bound state, solutions such as shining a light pulse using a light emitting diode to create excess electron-hole pairs or using an ohmic path to controllably inject electrons can be employed.

Appendix E Qubit operation

The qubit operation parameters of the single-atom device presented in this paper are adopted from a commonly used cQED approach Blais et al. (2004). To estimate the initialization, manipulation and readout performance, we focus on the Nb/P-2S configuration with g/2π=0.24 MHz, a realistic resonator internal quality factor Qi=4×104 and an external quality factor Qe=4×105 easily realizable according to our estimation of capacitive coupling Martinis et al. (2014), or galvanic coupling. We also assume that the resonator frequency and the Zeeman splitting frequency are ω0/2π=5.6 GHz and ωs/2π≈5.6 GHz, respectively.

The zero-detuning (Δ≡ωs−ω0=0) relaxation time limited by the Purcell effect for this strongly-coupled system is obtained as T1,init=Γ−1P=2κ−1=2.3μs Sete et al. (2014) which is several orders of magnitude shorter than the free spin relaxation. Effective initialization of spin systems using resonator coupling has been previously observed Bienfait et al. (2016), and the 2-order of magnitude increase in g in our device promises a much faster initialization. The linear dependence of ΓP on κ is a result of strong coupling, which distinguishes it from previously measured weakly coupled systems Bienfait et al. (2016) where ΓP∝κ−1.

For spin manipulation, we can Stark shift the spin states using the electrodes shown in Figs. 1(b)-(d), or use magnetic field tuning the perturb the Zeeman energy. For a demonstration of the qubit operation parameters of the device, we chose to operate at Δ=40g, where the strong-coupling Purcell rate becomes ΓP/2π=14π(κ−√2√−A+√A2+κ2Δ2)=88 Hz with A≡Δ2+4g2−κ2/4Sete et al. (2014), giving rise to T2,rot=2/ΓP=1.8 ms. This relaxation time is not far from the measured Hahn-Echo TH2=1.1 ms for a P donor electron spin in enriched 28Si, believed to be limited primarily by the static magnetic field noise and thermal noise and not due to the proximity to the oxide layers or other amorphous material Muhonen et al. (2014). Therefore, it is reasonable to assume that the spin T2 time in our device is Purcell limited, hence set by T2,rot. Note that for the Al/Bi set, the direction of the Stark shift must be such that the |F,mF⟩=|5,−5⟩↔|4,−4⟩ transition frequency, already reduced by a relatively small B0 to lift the multiplet degeneracy, is further reduced to avoid exciting the higher frequency multiplet transitions. In this detuned qubit control regime, the microwave drive frequency is ωμw/2π=ωs/2π+(2nlim+1)g2/2πΔ=5.614406 GHz where photon number nlim=Δ2/4g2=400 sets the maximum drive power Blais et al. (2004), and is lower than the critical photon number ncrit≈1800 corresponding to the spiral inductor’s critical current. The qubit rotation speed, at on-resonance photon number nres≈17×106 set by nlim and ωμw, is frot=gκ√nres/πΔ=29 MHz Haroche (1993), corresponding to Nπ=2frotT2,rot>105 coherent π-rotations.

The spin readout also occurs in the same detuned regime, where cavity frequency is “pulled” giving rise to a fsep=g2/πΔ=12 kHz separation frequency depending on the spin state. This corresponds to a phase shift of ϕ=2arctan(2g2/κΔ)=10∘, well above the measurement sensitivity usually considered to be 0.1∘, and suggests an extremely high-fidelity readout. The measurement time to resolve ϕ is estimated to be Tm=(2κNreadθ20)−1=3.1μs where θ0≡2g2/κΔBlais et al. (2004) and Nread=25 is the readout photon number (one must use Nread<Δ2/4g2).

DISCLAIMER: Certain commercial equipment, instruments, or materials (or suppliers, or software, …) are identified in this paper to foster understanding. Such identification does not imply recommendation or endorsement by the National Institute of Standards and Technology, nor does it imply that the materials or equipment identified are necessarily the best available for the purpose.

See Supplemental Material at [URL will be
inserted by publisher] for the details of inductance simulations and
calculations, coupling rate calculations, estimation of the current density
and resonator quality factor, possibility of donor ionization, and qubit
operation.

See Supplemental Material at [URL will be
inserted by publisher] for the details of inductance simulations and
calculations, coupling rate calculations, estimation of the current density
and resonator quality factor, possibility of donor ionization, and qubit
operation.

See Supplemental Material at [URL will be
inserted by publisher] for the details of inductance simulations and
calculations, coupling rate calculations, estimation of the current density
and resonator quality factor, possibility of donor ionization, and qubit
operation.

See Supplemental Material at [URL will be
inserted by publisher] for the details of inductance simulations and
calculations, coupling rate calculations, estimation of the current density
and resonator quality factor, possibility of donor ionization, and qubit
operation.

See Supplemental Material at [URL will be
inserted by publisher] for the details of inductance simulations and
calculations, coupling rate calculations, estimation of the current density
and resonator quality factor, possibility of donor ionization, and qubit
operation.

See Supplemental Material at [URL will be
inserted by publisher] for the details of inductance simulations and
calculations, coupling rate calculations, estimation of the current density
and resonator quality factor, possibility of donor ionization, and qubit
operation.